1. Field of the Invention
This invention relates generally to seismic data processing and more particularly to a method for compensating for the effects of irregular spatial data sampling of seismic wavefields.
2. Discussion of the Prior Art
In the art of seismic exploration, an acoustic wavefield (a shot) is generated by an acoustic source. The wavefield propagates through the earth from a source location. The wavefield is reflected from earth layers beneath the surface whence it returns to the surface. A plurality of seismic detectors are distributed on or near the surface of the earth, remotely from the source location, along lines of survey or in large areal patches. Preferably, the detectors, which constitute discrete wavefield sampling stations, are uniformly distributed spatially so that the wavefield can be uniformly sampled both areally and temporally. The sampled data are quantized and archivally recorded for further processing.
From a practical standpoint, the ideal uniform distribution is sometimes not possible because of obstructions such as buildings, roads or other culture. In the case of marine exploration, the distribution is irregular because of errors in the assumed detector locations due to cable drift because of currents and wind or due to the presence of drilling and production platforms. The seismic data are often not only locally under-sampled, they also may be locally excessively densely sampled.
Well-known seismic data-processing algorithms such as stacking, multi-channel filtering, dip moveout correction (DMO), prestack migration, velocity analyses, anisotropy studies, migration and wavefield extrapolation, all assume that the data are uniformly sampled. However, in operation, the data gathered may be irregularly sampled. That irregularity may be due to obstructions as earlier explained or to missing shots or to inoperative detectors or receivers. When such irregular or inadequately spatially sampled data are not corrected, unwanted computational artifacts may result that are superimposed upon the processed output data. Wave-equation processing routines such as DMO and movement of data samples, such as by repositioning because of dip migration, can be affected by irregular spatial sampling. In fact, the algorithm meant to improve the output data actually degrades that data because spatial under-sampling or over-sampling leaves remanents of the data-processing operators in the output data. Gross under-sampling, of course may result in dip aliasing. Noise, which is offset-dependent, is only partly canceled out in the stacking operation in the presence of locally-sparse spatial sampling.
In this disclosure, the term "operator" will be used frequently. The term is defined to mean a specific thing involved in a data-processing operation. Thus a DMO operator is a specific expression involved in applying a correction to normal moveout for dip. An operator may be expressed as a symbol indicating an operation to be performed and itself may be the subject of mathematical manipulation.
Various authorities have addressed the problem of locally-sparse spatial wavefield sampling. In a paper entitled Wave-equation Trace Interpolation, (Geophysics, vol. 52, no. 7, July 1987, pp 973-984) J. Ronen discloses that a processing sequence in which one treats missing data as zero data and performs partial migration before stacking is equivalent to application of the transpose of the stacking operator that actually needs to be inverted. Ronen states that the inverse of the operator cannot be uniquely determined but it can be estimated using spatial spectral balancing in a conjugate-gradient iterative scheme. The first iteration is simply prestack partial migration. Where spatial aliasing is present, several additional iterations are needed.
R. G. Williams et al, in a paper delivered at the 51st annual meeting of the EAEG, May 1989, entitled Model-constrained Anti-alias Filtering for Improved DMO, apply an anti-alias filter to the data so that the azimuthal under-sampling of the DMO operator can be reduced by use of a dip model for the survey to limit the operator aperture in an azimuth-dependent manner.
In a paper published as expanded abstract no. 1144, at the 59th International Meeting of the SEG, 1989, entitled Effect of Irregular Sampling on Prestack DMO, J. Black et al. explain that irregular midpoint and azimuthal distributions of the seismic data will cause artifacts in the DMO output if those irregular distributions are ignored. That is said to be especially a problem for pre-stack DMO with land data-collection geometries. Using the actual data-collection geometry, the DMO response can be computed for flat events and can be used to design long wavelength corrections and selective edits that minimize the impact of DMO-induced artifacts. This procedure can be used with any DMO implementation but it is especially relevant for 3-D DMO using the Kirchhoff method.
Although the above references have addressed specific facets of the problems of locally sparse or irregular spatial sampling, there yet remains a need for a comprehensive, accurate and efficient solution to the problem that is computationally economical.